Moments

The $k-th$ moment of a distribution is defined as:

$$\langle x^k \rangle = \frac{1}{N} \sum_{i=1}^N x_i^k.$$ (262)

If the numbers are distributed with a uniform probability distribution $P(x)$, then (264) must correspond to the moment of $P$:

$$\int _0^1 {x^kP(x)dx} \sim \frac{1}{k+1}.$$

If this holds for your generator, then you know that the distribution is uniform. If the deviation from this varies as $1/\sqrt{N}$, then you also know that the distribution is random.